Current Electricity

When a voltage is applied across a conductor, electrons don't shoot through — they drift slowly while colliding with the lattice. The drift velocity is:

v_d = (eE/m) · τ = (eV / mL) · τ

where e is electron charge, m its mass, E the field, L the length, and τ the average time between collisions (relaxation time, ~10⁻¹⁴ s in copper).

Current in a conductor of cross-section A with n electrons per unit volume:

I = neAv_d

Combining: I = neA · (eEτ/m) = (ne²τ/m) · A · (V/L). Rearranging gives Ohm's law:

V = IR, where R = ρL/A and ρ = m/(ne²τ) is resistivity.

Key insight: drift velocity is tiny (~0.1 mm/s in a household wire) — yet bulbs light instantly because the electric field propagates at near light speed, pushing on every electron in the wire simultaneously.

Resistivity ρ depends on temperature (linearly for metals, near room temperature):
ρ_T = ρ_0 [1 + α(T − T_0)]

For metals, α > 0 (resistance grows with temperature). For semiconductors and electrolytes, α < 0 (resistance falls).

ELECTRIC CURRENT (I) = rate of flow of charge.
I = dQ/dt. Unit: ampere (A) = C/s.

Conventional current: direction of positive charge flow (opposite to electrons).

OHM'S LAW

V = IR (for ohmic conductors at constant temperature).

• Resistance (R): units ohm (Ω). R = V/I.
• Conductance (G) = 1/R. Unit: siemens (S).
• Ohmic: linear V-I. Non-ohmic: diode, transistor, electrolytes.

Resistivity (ρ): R = ρL/A.

• Material property. Cu: 1.7×10⁻⁸ Ω·m; Al: 2.8×10⁻⁸; Si: 6.4×10² (semiconductor); glass: 10¹⁰⁻¹⁴.
• Conductivity σ = 1/ρ.

Temperature dependence:

• Metals: ρ increases with T → R increases.
• Semiconductors: ρ decreases with T (more charge carriers).

DRIFT VELOCITY & CURRENT

• Free electrons in conductor move randomly; in field, slow drift superimposed.
• Drift velocity v_d = (eE/m) τ (τ = relaxation time, ~10⁻¹⁴ s).
• v_d ~ 10⁻⁴ m/s (very slow!). Signal travels near c because field propagates fast.

Current density J = I/A = neAv_d / A = nev_d.

Microscopic Ohm's law: J = σE.

COMBINATIONS OF RESISTORS

Series: R = R₁ + R₂ + R₃ + ...

• Same current through each.

Parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃ + ...

• Same voltage across each.

EMF AND INTERNAL RESISTANCE

• Battery EMF (ε) = potential when no current.
• Internal resistance (r): V_terminal = ε − Ir.
• During charging: V = ε + Ir.

KIRCHHOFF'S LAWS

1. Junction (current) rule: Σ I_in = Σ I_out (conservation of charge).

2. Loop (voltage) rule: Σ V around any closed loop = 0 (conservation of energy).

Sign convention: going with current, IR drop is negative. Going through EMF + to −, it's positive.

WHEATSTONE BRIDGE

Four resistors in a bridge with galvanometer in middle. Balanced when:

P/Q = R/S (no current through galvanometer).

Applications:

• Meter bridge (slide-wire bridge) for unknown R.
• Strain gauge circuits.

POTENTIOMETER

A long wire of uniform resistance. Used for:

• Comparing EMFs of two cells (no current drawn → exact EMF).
• Measuring internal resistance.
• Calibrating ammeters/voltmeters.

EMF measured by balance length: ε ∝ L.

POWER & ENERGY

• P = VI = I²R = V²/R. Unit: watt.
• Energy: W = Pt = VIt = I²Rt. Unit: joule. Commercial: kWh.
• 1 kWh = 3.6 × 10⁶ J.
• For domestic circuit: P = V²/R (V is constant 230 V).

EXAMPLE 1:
A wire of 10 Ω is cut into two equal halves. Resistance of each piece?

R = ρL/A. Length halves → R halves → each piece is 5 Ω.

EXAMPLE 2 (Wheatstone):
In a Wheatstone bridge, P = 10 Ω, Q = 20 Ω, R = 15 Ω. Find S at balance.

P/Q = R/S → 10/20 = 15/S → S = 30 Ω.

EXAMPLE 3 (Kirchhoff):
Two batteries 12 V (r=1Ω) and 6 V (r=2Ω) connected in parallel with same polarity, supplying a 4Ω load. Find current through load.

Use mesh or thevenin: V_th = (12/1 + 6/2)/(1/1 + 1/2) = 15 / 1.5 = 10 V. r_th = 1·2/(1+2) = 2/3 Ω.
I = 10/(4 + 2/3) = 10/(14/3) = 30/14 ≈ 2.14 A.

EXAM HOOKS:

• Drift velocity is tiny but field propagates at speed of light.
• Resistors in series: same I. In parallel: same V.
• Internal resistance reduces terminal voltage when current flows.
• Wheatstone balance: P/Q = R/S (no current in galvanometer).
• Potentiometer is more accurate than voltmeter (no current drawn).

Kirchhoff's current law (KCL): at any junction, total current in = total current out. Conservation of charge.

Kirchhoff's voltage law (KVL): around any closed loop, sum of EMFs equals sum of voltage drops. Conservation of energy.

The 5-step procedure for solving any DC circuit:

1. Mark currents in each branch (assume directions; signs sort out at the end).
2. Apply KCL at every junction except one (n junctions → n−1 independent equations).
3. Choose loops to cover every branch.
4. Apply KVL to each loop. Sign conventions:
   • Cross EMF from − to + → +EMF
   • Cross EMF from + to − → −EMF
   • Cross resistor in direction of current → −IR
   • Cross resistor opposite to current → +IR
5. Solve the linear system. Negative result for a current means you guessed the wrong direction — magnitude is still correct.

Worked example. Two batteries (12 V and 6 V) in a loop with two resistors (2 Ω in series with the 12V battery, 3 Ω in series with the 6V). Take loop current I clockwise. KVL: +12 − 2I − 6 − 3I = 0 → 5I = 6 → I = 1.2 A. The current is 1.2 A clockwise, confirming the 12 V battery dominates.

ELECTRIC CURRENT (I) = rate of flow of charge.
I = dQ/dt. Unit: ampere (A) = C/s.

Conventional current: direction of positive charge flow (opposite to electrons).

OHM'S LAW

V = IR (for ohmic conductors at constant temperature).

• Resistance (R): units ohm (Ω). R = V/I.
• Conductance (G) = 1/R. Unit: siemens (S).
• Ohmic: linear V-I. Non-ohmic: diode, transistor, electrolytes.

Resistivity (ρ): R = ρL/A.

• Material property. Cu: 1.7×10⁻⁸ Ω·m; Al: 2.8×10⁻⁸; Si: 6.4×10² (semiconductor); glass: 10¹⁰⁻¹⁴.
• Conductivity σ = 1/ρ.

Temperature dependence:

• Metals: ρ increases with T → R increases.
• Semiconductors: ρ decreases with T (more charge carriers).

DRIFT VELOCITY & CURRENT

• Free electrons in conductor move randomly; in field, slow drift superimposed.
• Drift velocity v_d = (eE/m) τ (τ = relaxation time, ~10⁻¹⁴ s).
• v_d ~ 10⁻⁴ m/s (very slow!). Signal travels near c because field propagates fast.

Current density J = I/A = neAv_d / A = nev_d.

Microscopic Ohm's law: J = σE.

COMBINATIONS OF RESISTORS

Series: R = R₁ + R₂ + R₃ + ...

• Same current through each.

Parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃ + ...

• Same voltage across each.

EMF AND INTERNAL RESISTANCE

• Battery EMF (ε) = potential when no current.
• Internal resistance (r): V_terminal = ε − Ir.
• During charging: V = ε + Ir.

KIRCHHOFF'S LAWS

1. Junction (current) rule: Σ I_in = Σ I_out (conservation of charge).

2. Loop (voltage) rule: Σ V around any closed loop = 0 (conservation of energy).

Sign convention: going with current, IR drop is negative. Going through EMF + to −, it's positive.

WHEATSTONE BRIDGE

Four resistors in a bridge with galvanometer in middle. Balanced when:

P/Q = R/S (no current through galvanometer).

Applications:

• Meter bridge (slide-wire bridge) for unknown R.
• Strain gauge circuits.

POTENTIOMETER

A long wire of uniform resistance. Used for:

• Comparing EMFs of two cells (no current drawn → exact EMF).
• Measuring internal resistance.
• Calibrating ammeters/voltmeters.

EMF measured by balance length: ε ∝ L.

POWER & ENERGY

• P = VI = I²R = V²/R. Unit: watt.
• Energy: W = Pt = VIt = I²Rt. Unit: joule. Commercial: kWh.
• 1 kWh = 3.6 × 10⁶ J.
• For domestic circuit: P = V²/R (V is constant 230 V).

EXAMPLE 1:
A wire of 10 Ω is cut into two equal halves. Resistance of each piece?

R = ρL/A. Length halves → R halves → each piece is 5 Ω.

EXAMPLE 2 (Wheatstone):
In a Wheatstone bridge, P = 10 Ω, Q = 20 Ω, R = 15 Ω. Find S at balance.

P/Q = R/S → 10/20 = 15/S → S = 30 Ω.

EXAMPLE 3 (Kirchhoff):
Two batteries 12 V (r=1Ω) and 6 V (r=2Ω) connected in parallel with same polarity, supplying a 4Ω load. Find current through load.

Use mesh or thevenin: V_th = (12/1 + 6/2)/(1/1 + 1/2) = 15 / 1.5 = 10 V. r_th = 1·2/(1+2) = 2/3 Ω.
I = 10/(4 + 2/3) = 10/(14/3) = 30/14 ≈ 2.14 A.

EXAM HOOKS:

• Drift velocity is tiny but field propagates at speed of light.
• Resistors in series: same I. In parallel: same V.
• Internal resistance reduces terminal voltage when current flows.
• Wheatstone balance: P/Q = R/S (no current in galvanometer).
• Potentiometer is more accurate than voltmeter (no current drawn).

ELECTRIC CURRENT (I) = rate of flow of charge.
I = dQ/dt. Unit: ampere (A) = C/s.

Conventional current: direction of positive charge flow (opposite to electrons).

OHM'S LAW

V = IR (for ohmic conductors at constant temperature).

• Resistance (R): units ohm (Ω). R = V/I.
• Conductance (G) = 1/R. Unit: siemens (S).
• Ohmic: linear V-I. Non-ohmic: diode, transistor, electrolytes.

Resistivity (ρ): R = ρL/A.

• Material property. Cu: 1.7×10⁻⁸ Ω·m; Al: 2.8×10⁻⁸; Si: 6.4×10² (semiconductor); glass: 10¹⁰⁻¹⁴.
• Conductivity σ = 1/ρ.

Temperature dependence:

• Metals: ρ increases with T → R increases.
• Semiconductors: ρ decreases with T (more charge carriers).

DRIFT VELOCITY & CURRENT

• Free electrons in conductor move randomly; in field, slow drift superimposed.
• Drift velocity v_d = (eE/m) τ (τ = relaxation time, ~10⁻¹⁴ s).
• v_d ~ 10⁻⁴ m/s (very slow!). Signal travels near c because field propagates fast.

Current density J = I/A = neAv_d / A = nev_d.

Microscopic Ohm's law: J = σE.

COMBINATIONS OF RESISTORS

Series: R = R₁ + R₂ + R₃ + ...

• Same current through each.

Parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃ + ...

• Same voltage across each.

EMF AND INTERNAL RESISTANCE

• Battery EMF (ε) = potential when no current.
• Internal resistance (r): V_terminal = ε − Ir.
• During charging: V = ε + Ir.

KIRCHHOFF'S LAWS

1. Junction (current) rule: Σ I_in = Σ I_out (conservation of charge).

2. Loop (voltage) rule: Σ V around any closed loop = 0 (conservation of energy).

Sign convention: going with current, IR drop is negative. Going through EMF + to −, it's positive.

WHEATSTONE BRIDGE

Four resistors in a bridge with galvanometer in middle. Balanced when:

P/Q = R/S (no current through galvanometer).

Applications:

• Meter bridge (slide-wire bridge) for unknown R.
• Strain gauge circuits.

POTENTIOMETER

A long wire of uniform resistance. Used for:

• Comparing EMFs of two cells (no current drawn → exact EMF).
• Measuring internal resistance.
• Calibrating ammeters/voltmeters.

EMF measured by balance length: ε ∝ L.

POWER & ENERGY

• P = VI = I²R = V²/R. Unit: watt.
• Energy: W = Pt = VIt = I²Rt. Unit: joule. Commercial: kWh.
• 1 kWh = 3.6 × 10⁶ J.
• For domestic circuit: P = V²/R (V is constant 230 V).

EXAMPLE 1:
A wire of 10 Ω is cut into two equal halves. Resistance of each piece?

R = ρL/A. Length halves → R halves → each piece is 5 Ω.

EXAMPLE 2 (Wheatstone):
In a Wheatstone bridge, P = 10 Ω, Q = 20 Ω, R = 15 Ω. Find S at balance.

P/Q = R/S → 10/20 = 15/S → S = 30 Ω.

EXAMPLE 3 (Kirchhoff):
Two batteries 12 V (r=1Ω) and 6 V (r=2Ω) connected in parallel with same polarity, supplying a 4Ω load. Find current through load.

Use mesh or thevenin: V_th = (12/1 + 6/2)/(1/1 + 1/2) = 15 / 1.5 = 10 V. r_th = 1·2/(1+2) = 2/3 Ω.
I = 10/(4 + 2/3) = 10/(14/3) = 30/14 ≈ 2.14 A.

EXAM HOOKS:

• Drift velocity is tiny but field propagates at speed of light.
• Resistors in series: same I. In parallel: same V.
• Internal resistance reduces terminal voltage when current flows.
• Wheatstone balance: P/Q = R/S (no current in galvanometer).
• Potentiometer is more accurate than voltmeter (no current drawn).

ELECTRIC CURRENT (I) = rate of flow of charge.
I = dQ/dt. Unit: ampere (A) = C/s.

Conventional current: direction of positive charge flow (opposite to electrons).

OHM'S LAW

V = IR (for ohmic conductors at constant temperature).

• Resistance (R): units ohm (Ω). R = V/I.
• Conductance (G) = 1/R. Unit: siemens (S).
• Ohmic: linear V-I. Non-ohmic: diode, transistor, electrolytes.

Resistivity (ρ): R = ρL/A.

• Material property. Cu: 1.7×10⁻⁸ Ω·m; Al: 2.8×10⁻⁸; Si: 6.4×10² (semiconductor); glass: 10¹⁰⁻¹⁴.
• Conductivity σ = 1/ρ.

Temperature dependence:

• Metals: ρ increases with T → R increases.
• Semiconductors: ρ decreases with T (more charge carriers).

DRIFT VELOCITY & CURRENT

• Free electrons in conductor move randomly; in field, slow drift superimposed.
• Drift velocity v_d = (eE/m) τ (τ = relaxation time, ~10⁻¹⁴ s).
• v_d ~ 10⁻⁴ m/s (very slow!). Signal travels near c because field propagates fast.

Current density J = I/A = neAv_d / A = nev_d.

Microscopic Ohm's law: J = σE.

COMBINATIONS OF RESISTORS

Series: R = R₁ + R₂ + R₃ + ...

• Same current through each.

Parallel: 1/R = 1/R₁ + 1/R₂ + 1/R₃ + ...

• Same voltage across each.

EMF AND INTERNAL RESISTANCE

• Battery EMF (ε) = potential when no current.
• Internal resistance (r): V_terminal = ε − Ir.
• During charging: V = ε + Ir.

KIRCHHOFF'S LAWS

1. Junction (current) rule: Σ I_in = Σ I_out (conservation of charge).

2. Loop (voltage) rule: Σ V around any closed loop = 0 (conservation of energy).

Sign convention: going with current, IR drop is negative. Going through EMF + to −, it's positive.

WHEATSTONE BRIDGE

Four resistors in a bridge with galvanometer in middle. Balanced when:

P/Q = R/S (no current through galvanometer).

Applications:

• Meter bridge (slide-wire bridge) for unknown R.
• Strain gauge circuits.

POTENTIOMETER

A long wire of uniform resistance. Used for:

• Comparing EMFs of two cells (no current drawn → exact EMF).
• Measuring internal resistance.
• Calibrating ammeters/voltmeters.

EMF measured by balance length: ε ∝ L.

POWER & ENERGY

• P = VI = I²R = V²/R. Unit: watt.
• Energy: W = Pt = VIt = I²Rt. Unit: joule. Commercial: kWh.
• 1 kWh = 3.6 × 10⁶ J.
• For domestic circuit: P = V²/R (V is constant 230 V).

EXAMPLE 1:
A wire of 10 Ω is cut into two equal halves. Resistance of each piece?

R = ρL/A. Length halves → R halves → each piece is 5 Ω.

EXAMPLE 2 (Wheatstone):
In a Wheatstone bridge, P = 10 Ω, Q = 20 Ω, R = 15 Ω. Find S at balance.

P/Q = R/S → 10/20 = 15/S → S = 30 Ω.

EXAMPLE 3 (Kirchhoff):
Two batteries 12 V (r=1Ω) and 6 V (r=2Ω) connected in parallel with same polarity, supplying a 4Ω load. Find current through load.

Use mesh or thevenin: V_th = (12/1 + 6/2)/(1/1 + 1/2) = 15 / 1.5 = 10 V. r_th = 1·2/(1+2) = 2/3 Ω.
I = 10/(4 + 2/3) = 10/(14/3) = 30/14 ≈ 2.14 A.

EXAM HOOKS:

• Drift velocity is tiny but field propagates at speed of light.
• Resistors in series: same I. In parallel: same V.
• Internal resistance reduces terminal voltage when current flows.
• Wheatstone balance: P/Q = R/S (no current in galvanometer).
• Potentiometer is more accurate than voltmeter (no current drawn).